I took my dead Time Capsule to an Apple Store and had a Genius verify that it was truly dead. The Genius wrote up a repair record that stated that it was truly dead and gave me an R number.
I then left the Apple Store and called AppleCare and explained the situation and gave them the R number provided by the Genius. Apple Care was happy to replace the Time Capsule (no questions asked) via RMA for free even though my Apple Care for my MacBook Air wasn't bought at the same time that the Time Capsule was purchased. They seem to understand that there is a power supply defect in their Time Capsules.
Within 2 days of the failure, I was back in business.
Luckily, I had backed-up my sparsebundles a few months prior, so I only lost a couple of months of backups, but still had the oldest backups.